A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Amp\`{e}re (MA) equation, $\det \nabla \psi(x,t) = \mathsf{G} \circ\psi (x,t)$. The novelty of our approach is a new so-called \emph{perturbation formulation} of MA, which constructs the solution map $\psi$ via composition of a sequence of near identity deformations of a uniform reference mesh. This allows us to utilize a simple, fast, and high order accurate implementation of the deformation method. We design SAM to satisfy both internal and external consistency requirements between stability, accuracy, and efficiency constraints, and show that the scheme is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers, while running approximately 50-100 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2$D$ gas dynamics. Specifically, we implement the $C$-method and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.